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A new approach to the asymptotics of Sobolev type orthogonal polynomials. (English) Zbl 1219.33006
The aim of the paper is to obtain Mehler-Heine type asymptotic formulas for the sequence of Sobolev orthogonal polynomials when the measure which appears in the continuous part is Laguerre or generalized Hermite. The key to obtain results is the transformation of the Sobolev orthogonality into a standard quasi-orthogonality. Some of the obtained results generalize results given in a paper written by the same authors [Asymptotic Anal. 66, No. 2, 103–117 (2010; Zbl 1190.33010)] and in one by R. Alvarez-Nodarse and J. J. Moreno-Balcázar [Indag. Math., New Ser. 15, No. 2, 151–165 (2004; Zbl 1064.41022)] and solve conjectures posed there and in a paper of H. Dueñas and F. Marcellán [J. Approx. Theory 162, No. 2, 421–440 (2010; Zbl 1190.33013)].

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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